Given a field 
and an extension field 
, an element 
is called algebraic over 
if it is a root of some nonzero polynomial with coefficients
in 
.
Obviously, every element of 
is algebraic over 
. Moreover, the sum, difference, product, and quotient of algebraic
elements are again algebraic. It follows that the simple extension
field 
is an algebraic extension of 
iff 
is algebraic over 
.
The imaginary unit i is algebraic over the field 
of real numbers since it is a root of the polynomial 
. Because its coefficients are integers,
it is even true that 
is algebraic over the field 
of rational numbers, i.e., it is an algebraic
number (and also an algebraic integer).
As a consequence, 
and 
are algebraic extensions of 
and 
respectively. (Here, 
is the complex field 
, whereas 
is the total ring
of fractions of the ring of Gaussian integers ![Z[i]](/images/equations/AlgebraicElement/Inline23.svg)
.)
